Lux is more of a bar than a restaurant, but they do serve a limited selection of mostly upscale food. It's fairly dark and the tables are generally small, with an oil lamp on each one - the kind under a glass cover. While most of the tables are in the central area, there are a few booths around the edges and these tend to be rather private, good places for conversations one doesn't want overheard. There is a back room, but only when he wants there to be, so it's not something that will always apply and customers aren't welcome there without an explicit invitation.

There's a raised area with a piano that Lucifer usually plays - he prefers classical pieces or ones he has composed himself, but his repertoire isn't limited to that. Whether he takes requests or not depends on his mood, but he doesn't allow anyone else to play.

In addition to a full bar, Lux is the place to go to for expensive and even famous vintages of brandy, wine, scotch, etc. Lucifer is fond of that sort of thing, so naturally he stocks them. He can also cater to unusual requests, though for that, you might have to speak directly to him instead of the bartenders.

Also, if your character is magically sensitive, the place practically glows with magic. It's larger on the inside than the outside and he's prone to redecorating with magic. There are quite a few protective spells on the place and these are particularly noticeable because he used his blood for them, so they pretty much declare that this is his territory to those who can see that kind of thing (you might also get an idea of vast power from them).

Have some image links (lazily not cropped - these are from issues 5 & 6 of Lucifer):

bar area
main area when crowded
main room empty